A Symmetric Galerkin Boundary Element Method for Linear Poroelasticity
2
PAMM · Proc. Appl. Math. Mech. 11, 247 – 248 (2011) / DOI 10.1002/pamm.201110115 A Symmetric Galerkin Boundary Element Method for Linear Poroelasticity Michael Messner 1, * and Martin Schanz 1, ** 1 Institute of Applied Mechanics, Graz University of Technology In linear poroelasticity so far only collocation boundary element methods have been available. However, in some applications, e.g., when coupling with finite elements is desired, a symmetric formulation is preferable. Choosing a Galerkin approach which involves the second boundary integral equation, such a formulation is possible. Here, a previously presented integration by part technique for the regularization of the first boundary integral equation is extended to the second boundary integral equation as well. While the weakly singular representation of the double layer operator has been presented before, the emphasis lies here on the so called hyper-singular boundary integral operator. Due to the regularization, this operator can be evaluated numerically and, hence, be used within a numerical scheme for the first time. Different numerical studies will be presented to show the behavior of the established symmetric Galerkin boundary element method, also comparing it with collocation boundary element methods. c 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction A symmetric Galerkin boundary element method for the simulation of linear poroelastic media is presented. The first and second boundary integral equations related to Biot’s theory [1] are derived and used for the description of a mixed boundary value problem [2]. Applying the reformulated Convolution Quadrature Method [3] for the time discretization leads to a set of decoupled Laplace domain problems to be solved. Finally their solution is obtained by application of a Galerkin scheme [2] leading to a set of linear systems ˆ V DD - ˆ K DN ˆ K 0 ND ˆ D NN ˆ t D ˆ u N = ˆ f D ˆ f N e.g. ˆ K DN [i, j ] = h ˆ Kϕ j ,ψ i i = Z ψ i Z ˆ T * x ˆ U * > ϕ j dΓ j dΓ i ˆ D NN [i, j ] = h ˆ Dϕ j ,ϕ i i = Z ϕ i ˆ T x Z ˆ T * y ˆ U * > ϕ j dΓ j dΓ i , (1) where ˆ U * is the Laplace domain fundamental solution of the adjoint problem and ˆ T * the appropriate traction operator, moreover ˆ T is the traction operator of the problem itself. However, the bilinear forms as given above are not suitable for numerical treatment due to the singular behaviour of the integral’s kernel functions. The regularization of the double layer was presented in [6] and has been extended to the hyper-singular boundary integral operator for the present formulation. 2 Regularization of the Hyper-singular Operator For this section the linear elastic fundamental solution ˆ U E and the related traction operator ˆ T E reflecting Hooke’s law are introduced and similar for the Laplace equation with ˆ P L and ∂ ∂n . Both these problems are selfadjoint, thus one can forget about () * and () > . Moreover, splitting the unknown field ˆ u into the solid displacement ˆ u s and fluid pressure ˆ p f , the hyper-singular operator from (1) can be expanded in a series in terms of the distance r = |x - y| h ˆ D ˆ u, vi = Z v > " ˆ T E x Z ˆ T E y ˆ U E ˆ u s ds y ˆ T E x Z ˆ U E n y ˆ p f ds y n > x Z ˆ T E y ˆ U E ˆ u s ds y ∂ ∂nx Z ∂ ∂ny ˆ P L ˆ p f ds y # ds x + Z v > Z O ( r 0 ) ˆ u ds y ds x . (2) Above, one can observe that the same regularization techniques as for linear elasticity [4] and Laplace equation can be applied, i.e. integrating the first integral by parts, the derivatives acting on the fundamental solution can be shifted to the ansatz and test functions, respectively. This finally leads to a realization of the hyper-singular integral operator with only a weakly singular kernel function. * Email [email protected], phone +43 316 873 4519, fax +43 316 873 7641 ** Email [email protected], phone +43 316 873 7600, fax +43 316 873 7641 c 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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Transcript of A Symmetric Galerkin Boundary Element Method for Linear Poroelasticity
PAMM · Proc. Appl. Math. Mech. 11, 247 – 248 (2011) / DOI 10.1002/pamm.201110115
A Symmetric Galerkin Boundary Element Method for LinearPoroelasticity
Michael Messner1,∗ and Martin Schanz1,∗∗
1 Institute of Applied Mechanics, Graz University of Technology
In linear poroelasticity so far only collocation boundary element methods have been available. However, in some applications,e.g., when coupling with finite elements is desired, a symmetric formulation is preferable. Choosing a Galerkin approachwhich involves the second boundary integral equation, such a formulation is possible. Here, a previously presented integrationby part technique for the regularization of the first boundary integral equation is extended to the second boundary integralequation as well. While the weakly singular representation of the double layer operator has been presented before, theemphasis lies here on the so called hyper-singular boundary integral operator. Due to the regularization, this operator canbe evaluated numerically and, hence, be used within a numerical scheme for the first time. Different numerical studies willbe presented to show the behavior of the established symmetric Galerkin boundary element method, also comparing it withcollocation boundary element methods.
c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
A symmetric Galerkin boundary element method for the simulation of linear poroelastic media is presented. The first andsecond boundary integral equations related to Biot’s theory [1] are derived and used for the description of a mixed boundaryvalue problem [2]. Applying the reformulated Convolution Quadrature Method [3] for the time discretization leads to a set ofdecoupled Laplace domain problems to be solved. Finally their solution is obtained by application of a Galerkin scheme [2]leading to a set of linear systems
[VDD −KDN
K′ND DNN
] [tDuN
]=
[fDfN
]e.g.
KDN [i, j] = 〈Kϕj , ψi〉 =
∫ψi
∫ (T ∗x U∗
)>ϕjdΓjdΓi
DNN [i, j] = 〈Dϕj , ϕi〉 =
∫ϕiTx
∫ (T ∗y U∗
)>ϕjdΓjdΓi ,
(1)
where U∗ is the Laplace domain fundamental solution of the adjoint problem and T ∗ the appropriate traction operator,moreover T is the traction operator of the problem itself. However, the bilinear forms as given above are not suitable fornumerical treatment due to the singular behaviour of the integral’s kernel functions. The regularization of the double layerwas presented in [6] and has been extended to the hyper-singular boundary integral operator for the present formulation.
2 Regularization of the Hyper-singular Operator
For this section the linear elastic fundamental solution UE and the related traction operator T E reflecting Hooke’s law areintroduced and similar for the Laplace equation with PL and ∂
∂n . Both these problems are selfadjoint, thus one can forget about( )∗ and ( )
>. Moreover, splitting the unknown field u into the solid displacement us and fluid pressure pf , the hyper-singularoperator from (1) can be expanded in a series in terms of the distance r = |x− y|
〈Du,v〉 =
∫v>
[ T Ex
∫ (T Ey UE
)usdsy T E
x
∫ (UEny
)pfdsy
n>x
∫ (T Ey UE
)usdsy ∂
∂nx
∫ (∂
∂nyPL)pf dsy
]dsx +
∫v>∫
O(r0)u dsydsx . (2)
Above, one can observe that the same regularization techniques as for linear elasticity [4] and Laplace equation can be applied,i.e. integrating the first integral by parts, the derivatives acting on the fundamental solution can be shifted to the ansatz and testfunctions, respectively. This finally leads to a realization of the hyper-singular integral operator with only a weakly singularkernel function.
∗ Email [email protected], phone +43 316 873 4519, fax +43 316 873 7641∗∗ Email [email protected], phone +43 316 873 7600, fax +43 316 873 7641
c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
248 Section 4: Structural mechanics
3 Numerical Example - One Dimensional Wave PropagationOne dimensional wave propagation is simulated with the symmetric Galerkin BEM(SGBEM) and compared in Fig.1 with the semi-analytic solution (poro1D) [5] and thepreviously presented collocation BEM (CBEM) [6]. For this purpose a poroelastic col-umn of 3[m] length with a cross-sectional area of 1[m2] is considered. This columnis loaded by a unit-step stress load of −1[N/m2] at time t > 0 and zero pressure atall times on the front face, while all other faces are supported by impermeable rollerbearings, i.e. zero in plane stresses and out of plane displacement as well as flux. Sup-plementary to these constraints the Poisson’s ratio of berea sandstone [7] is set to zerofor better agreement with the one dimensional reference solution.
roller
beari
ngs
x1
x2
x3
p = 0N/m2
t3 = −1N/m2
0 0.002 0.004 0.006 0.008 0.01time [sec]
-2.5e-10
-2e-10
-1.5e-10
-1e-10
-5e-11
0
disp
lace
men
t u 3 [
m]
poro1D
SGBEM 224SGBEM 896SGBEM 2016 CBEM 224 CBEM 896CBEM 2016
a)0.002 0.004 0.006 0.008 0.01
time [sec]
-0.5
0
0.5
1
1.5
2
2.5
stre
ss t 3 [
N/m
2 ] poro1D
SGBEM 224SGBEM 896SGBEM 2016CBEM 224CBEM 896CBEM 2016
b)
0 0.002 0.004 0.006 0.008 0.01time [sec]
-0.25
0
0.25
0.5
0.75
1
pres
sure
p [
N/m
2 ]
poro1D
SGBEM 224SGBEM 896SGBEM 2016CBEM 224CBEM 896CBEM 2016
c)0 0.002 0.004 0.006 0.008 0.01
time [sec]
0
7.5e-10
1.5e-09
2.25e-09
3e-09
flux
q [
m/s
ec] poro1D
SGBEM 224SGBEM 896SGBEM 2016CBEM 224CBEM 896CBEM 2016
d)
Fig. 1 Poroelastic results ploted over time: Solid displacement a, stress b, fluid pressure c and flux d are compared for different methods(SGBEM, CBEM) on different meshes (with 224, 896, 2016 triangles) with the reference solution (poro1D).
For the results in Fig.1 three different uniform discretizations with 224, 896 and 2016 linear triangles have been used. Onthese different meshes the solid displacement and fluid pressure were appoximated by piecewise linear continuous ansatzfunctions and the solid stresses and fluid flux by piecewise constants. Furthermore, for the time discretization a constantβCFL = c∆t
h of 0.3 was chosen to obtain reasonable results (c initial fast compressional wave speed). One can observe that theSGBEM performs better compared to the CBEM especially for coarser spatial discretizations, where it shows less damping.This advantage becomes more obvious the further in time one observes the results, see Fig.1 a - d.
Acknowledgements The authors gratefully acknowledge the financial support of the Austrian Science Fund (FWF) within the Dok-toratskolleg W1208 – Numerical Simulations in Technical Science.
References[1] M. A. Biot, J. Appl. Phys. 28, 168-191 (1956).[2] O. Steinbach and P. Urthaler, Bericht Inst. Num. Math. 3 (2008)[3] C. Lubich, Numer. Math. 52, 129-145/413-425 (1988).[4] H. Han, Numer. Math. 68, 269-281 (1994).[5] M. Schanz, and A. H. D. Cheng, Acta Mech. 145, 1-18 (2000).[6] M. Messner, and M. Schanz, Comput. Mech. online (2011).[7] Y. K. Kim and H. B. Kingsbury, Exp. Mech. 19, 252-258 (1979).
c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com
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